Kinematic Geometry of Gearing E-Book

David B. Dooner

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 718

Veröffentlichungsjahr: 2012

Beschreibung

Building on the first edition published in 1995 this new edition of Kinematic Geometry of Gearing has been extensively revised and updated with new and original material. This includes the methodology for general tooth forms, radius of torsure’, cylinder of osculation, and cylindroid of torsure; the author has also completely reworked the ‘3 laws of gearing’, the first law re-written to better parallel the existing ‘Law of Gearing” as pioneered by Leonard Euler, expanded from Euler’s original law to encompass non-circular gears and hypoid gears, the 2nd law of gearing describing a unique relation between gear sizes, and the 3rd law completely reworked from its original form to uniquely describe a limiting condition on curvature between gear teeth, with new relations for gear efficiency are presented based on the kinematics of general toothed wheels in mesh.  There is also a completely new chapter on gear vibration load factor and impact.

Progressing from the fundamentals of geometry to construction of gear geometry and application, Kinematic Geometry of Gearing presents a generalized approach for the integrated design and manufacture of gear pairs, cams and all other types of toothed/motion/force transmission mechanisms using computer implementation based on algebraic geometry.

Leseprobe

Contents

Cover

Title Page

Copyright

Preface

Part One: Fundamental Principles of Toothed Bodies in Mesh

1: Introduction to the Kinematics of Gearing

1.1 Introduction

1.2 An Overview

1.3 Nomenclature and Terminology

1.4 Reference Systems

1.5 The Input/Output Relationship

1.6 Rigid Body Assumption

1.7 Mobility

1.8 Arhnold-Kennedy Instant Center Theorem

1.9 Euler-Savary Equation for Envelopes

1.10 Conjugate Motion Transmission

1.11 Contact Ratio

1.12 Backlash

1.13 Special Toothed Bodies

1.14 Noncylindrical Gearing

1.15 Noncircular Gears

1.16 Schematic Illustration of Gear Types

1.17 Mechanism Trains

1.18 Summary

Part Two: The Kinematic Geometry of Conjugate Motion in Space

2: Kinematic Geometry of Planar Gear Tooth Profiles

2.1 Introduction

2.2 A Unified Approach to Tooth Profile Synthesis

2.3 Tooth Forms Used for Conjugate Motion Transmission

2.4 Contact Ratio

2.5 Dimensionless Backlash

2.6 Rack Coordinates

2.7 Planar Gear Tooth Profile

2.8 Summary

3: Generalized Reference Coordinates for Spatial Gearing—the Cylindroidal Coordinates

3.1 Introduction

3.2 Cylindroidal Coordinates

3.3 Homogeneous Coordinates

3.4 Screw Operators

3.5 The Generalized Equivalence of the Pitch Point—the Screw Axis

3.6 The Generalized Pitch Surface—Axodes

3.7 The Generalized Transverse Surface

3.8 The Generalized Axial Surface

3.9 Summary

4: Differential Geometry

4.1 Introduction

4.2 The Curvature of a Spatial Curve

4.3 The Torsion of a Spatial Curve

4.4 The First Fundamental Form

4.5 The Second Fundamental Form

4.6 Principal Directions and Principal Curvatures

4.7 Torsure of a Spatial Curve

4.8 The Cylindroid of Torsure

4.9 Ruled Surface Trihedrons

4.10 Formulas of Fernet-Serret

4.11 Coordinate Transformations

4.12 Characteristic Lines and Points

4.13 Summary

5: Analysis of Toothed Bodies for Motion Generation

5.1 Introduction

5.2 Spatial Mobility Criterion

5.3 Reciprocity—the First Law of Gearing

5.4 The Line Complex

5.5 The Tooth Spiral

5.6 Tooth Spiral Angle—the Second Law of Gearing

5.7 Reduced Tooth Curvature—the Third Law of Gearing

5.8 Classification of Gear Types

5.9 Contact Ratio

5.10 Spatial Backlash

5.11 Relative Displacements

5.12 Mesh Efficiency

5.13 Summary

6: The Manufacture of Toothed Bodies

6.1 Introduction

6.2 Manufacturing Background

6.3 Crossed Hyperboloidal Gears

6.4 Fabrication of Cutters

6.5 Gear Cutting Machine Layout

6.6 The Envelope of the Cutter

6.7 Material Removal Rate

6.8 Surface Cutting Speed

6.9 Discretization Error

6.10 Inspection

6.11 Hyperboloidal Blank Dimensions

6.12 Summary

7: Vibrations and Dynamic Loads in Gear Pairs

7.1 Introduction

7.2 Excitations

7.3 Transmission Error

7.4 Fourier Transforms

7.5 Impact Loading

7.6 Mesh Stiffness

7.7 Inertial Properties

7.8 Manufacturing Dynamics

7.9 Summary

Part Three: The Integrated Design and Manufacturing Process

8: Gear Design Rating

8.1 Introduction

8.2 Modes of Gear Failure

8.3 Reaction Loads

8.4 Gear Parameters for Specified Deflections

8.5 The Fillet Stress

8.6 Inertial Stress

8.7 Contact Stress

8.8 Minimum Film Thickness

8.9 Wear

8.10 Friction Coefficient

8.11 Flash Temperature

8.12 Thermal Stress

8.13 Failure Analysis

8.14 Windage Losses

8.15 Optimization

8.16 Summary

9: The Integrated CAD–CAM Process

9.1 Introduction

9.2 Modular Components for Geometric Synthesis

9.3 The Integrated CAD–CAM Process

9.4 Illustrative Example

9.5 Summary

10: Case Illustrations of the Integrated CAD–CAM Process

10.1 Introduction

10.2 Case 1

10.3 Case 2

10.4 Case 3

10.5 Case 4

10.6 Case 5

10.7 Case 6

10.8 Case 7

10.9 Case 8

10.10 Case 9

10.11 Case 10

10.12 Case 11

10.13 Case 12

10.14 Case 13

10.15 Case 14

10.16 Case 15

10.17 Case 16

10.18 Case 17

10.19 Case 18

10.20 Case 19

10.21 Case 20

10.22 Case 21

10.23 Case 22

10.24 Summary

Appendix A: Differential Expressions

A.1 Derivatives of the Radius of the Axode

A.2 Derivatives of the Included Angles

A.3 Derivatives of the Generators

A.4 Derivatives of the Pitch of the Instantaneous Twist

A.5 Derivatives of the Parameter of Distribution

A.6 Derivatives of the Striction Curve

A.7 Manufacturing Expressions

A.8 Derivatives of the Transverse Curve

A.9 Derivatives of the Angle Between the Generator and the Transverse Curve

A.10 Derivatives of the Spiral Angle

A.11 Derivatives of the Input Trihedron of Reference

A.12 Derivatives of the Cutter Parameters

Appendix B: On the Notation and Operations

Appendix C: Noncircular Gears

C.1 Torque and Speed Fluctuations in Rotating Shafts

C.2 2-dof Mechanical Function Generator

C.3 Steering Mechanism

C.4 Continuously Variable Transmission

C.5 Geared Robotic Manipulators

C.6 Spatial Mechanism for Body Guidance

C.7 Nonworking Profile

C.8 Multiple Reductions

Appendix D: The Delgear© Software

D.1 Installation

Appendix E: Splines

E.1 Cubic Splines

E.2 Natural Splines

E.3 NURBS

Appendix F: Contact Stress

F.1 Introduction

F.2 Background

F.3 Material Properties

F.4 Surface Geometry

F.5 Contact Deformations

F.6 Contact Area

F.7 Comparison

Appendix G: Glossary of Terms

Appendix H: Equilibrium and Diffusion Equations

H.1 Equilbrium Equations

H.2 Diffusion Equation Formulation

H.3 Expressions

Appendix I: On the Base Curve of Planar Noncircular Gears

Appendix J: Spatial Euler-Savary Equations

J.1 Planar Euler-Savary Equations

J.2 Hyperboloid of Osculation

J.3 Spatial Euler-Savary Equations

References

Index

This edition first published 2012 © 2012 John Wiley & Sons, Ltd

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Dooner, David B.

Kinematic geometry of gearing / David Dooner. – 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-119-95094-3 (hardback)

1. Gearing. 2. Machinery, Kinematics of. I. Title.

TJ184.D66 2012

621.8′33–dc23

2011050038

A catalogue record for this book is available from the British Library.

ISBN: 978-1-1199-5094-3

Preface

This second edition is an expansion of the first edition of The Kinematic Geometry of Gearing; A Concurrent Engineering Approach, introducing a generalized integrated methodology for the design and manufacture of different types of toothed bodies. Several expressions are modified from their original presentation along with a reorganization of the material. Included are changes in nomenclature to reduce subscripts, avoid conflicts with symbols, and aid in the implementation of computer software. The kinematic geometry of toothed bodies in mesh builds upon the original presentation and is supplemented with additional figures. The design and manufacturing sections are expanded to provide a more thorough evaluation of the new geometric methodology. Biographical data on historical individuals are provided in footnotes; much of this information is based on the work of J.J. O’Conner and E.F. Robertson.1 The ensuing presentation more thoroughly develops the single geometric methodology for integrated design and manufacture of gear pairs. A computer simulation for the integrated design and manufacture of generalized gear pairs has been completed (including a GUI or Graphical User Interface), showcasing the concurrent CAD–CAM of gear pairs. Prototype gear pairs have been fabricated and tested to illustrate the geometric methodology developed.

Two bodies in direct contact where the position and orientation of the output element are specified functions of a given input motion comprise gear pairs, threaded fasteners (i.e., bolts and nuts), as well as CAM systems. The overarching goal is a single geometric framework for the generalized design and manufacture of gear pairs, with consideration to fasteners and CAM systems. The importance of gearing continues in the twenty-first century. Gear elements range in size from 4000 mm or 150 in. to 1 μm or 0.5 μin., where the speeds range from less than 1 RPM to over 6 trillion RPM! The automotive industry is currently the biggest user of gear elements commanding over 60% of the world gear market. Such gears encompass spur and helical gears used in transmission, worm and worm wheels used for window regulation, along with spiral bevel and hypoid used in a rear/front axle assembly. Twenty percent of the automotive gear market is targeted to right angle drive gear pairs. It is estimated that there are over 800 million automobiles worldwide with 70 million produced annually. Gears are also essential machine elements in industrial applications, as well as the aerospace and marine industries.

Current gear practice for spatial gearing does not provide for bevel, hypoid, and worm gears to be treated with the same geometric considerations that are applicable to cylindrical gearing (namely, spur and helical gears). These geometric considerations include general formulations for the tooth profile, addendum and dedendum constants, profile modifications, crown, transverse and axial contact ratios, backlash, spiral angle variation, pressure angle variation, inspection techniques, as well as manufacturing technology. The salient theme of this book is to present a single geometric theory for the concurrent CAD–CAM of toothed bodies in direct contact used to transmit power (motion and load) between two axes. The end result is an axode-based theory analogous to that used to design and manufacture planar spur and helical gears. This unified approach is based on formulating a system of pitch, transverse, and axial surfaces, utilizing special curvilinear coordinates to parameterize the kinematic geometry of motion transmission between skew axes. Screw theory or the theory of screws is used as the basis for this geometric foundation. The same results can be obtained using alternatives such as dual numbers, Lie algebras, geometric algebras, or vector algebra. The presented technique builds upon existing known relations and utilizes screw theory to establish

cylindroidal coordinates,theorem of conjugate pitch surfaces,kinematic relations between generalized ruled surfaces,three laws of gearing,cylindroid of torsure, andspatial analog of planar Euler-Savary equations.

This analytical foundation is further expanded by introducing a variable diameter cutter or hob cutter for gear manufacture and developing the accompanying kinematic relations necessary for gear fabrication using the variable diameter cutter. A subtle facet of the entire integrated process for gear design and manufacture is the seamless integration of noncircular gear elements. Novel examples of noncircular gear pairs include

2-dof mechanical function generator (variable NC gear pair),spiral cylindrical and hypoid NC gear pairs,variable face width NC gear pairs,coiled NC gear pairs,coordinated automotive steering with NC gears,torque and speed balancing of rotating shafts, and1-dof mechanism for geared robotic manipulators.

AGMA published Gear Industry Vision in September 2004. The goal of this study was to define a vision for the gear community over the next 20 years where gears remain fundamental and the preferred solution in power transmission and control in the 2025 global marketplace. The final section of this study is Key Technological Challenges and Innovations. The top three objectives of this section are the following:

To establish a single system of design and testing standardsTo develop improved tribology modelingTo create predictive tools, virtual testing, and simulation tools

These objectives are central to this book. A single geometric system of design is presented by focusing on noncircular hyperboloidal gears for motion transmission between nonorthogonal axes. Although not immediately useful, such a system of design enables spur and helical, worm, and other forms to be readily obtained as a subset of noncircular hyperboloidal gears. About 15% ($5-8B) of the gear market targets rear axle assemblies consisting of spiral-bevel/hypoid gears. The basic differential gear train and spiral-bevel/hypoid gear set inherent in automotive axle assemblies has remained unchanged for over 75 years. Within the past 100 years, spiral-bevel/hypoid gear machine tool manufacturers have focused on a special fabricating process referred to as face milling and face hobbing. Inherent in this ``face’’ cutting process are limitations on the resulting end gear product. It is estimated that there exist over 15,000 spiral-bevel/hypoid gear cutting machines with over 25 years of life where the process presented in this book can eliminate some of these restrictions and lead to a new generation of gear fabrication.

As indicated, the central theme of this book is the presentation of a unified geometric methodology for the parameterization of gear pairs in mesh. An overall evaluation of the kinematic geometry of the newly synthesized gear pair is taken into account by including design rating formulas. These design rating formulas include fillet and inertial stress determination using finite-element analysis, contact stress, dynamics loads, wear, flash temperature, contact and bending fatigue analysis, reliability analysis, minimum lubricant film thickness, and specific film thickness, in addition to mesh and windage losses. An evaluation of the manufacturing process is performed by providing the cutting time, material removal rate, cutting power, surface cutting speed, and relative position and orientation between cutter and gear blank. This concurrent CAD–CAM methodology enables the designer to synthesize gear pairs with increased efficiency, reduced noise, while improving strength and surface durability. This development differs from current gear design and manufacturing practice.

The book is split into three parts and addresses both theory and practice. The first part revisits the concept of toothed bodies in mesh, their various forms for motion transmission, along with some terminology and nomenclature subsequently used to describe the concurrent design and manufacture of toothed bodies presented in Part Two. Part Two establishes the mathematical model used for the integrated design/manufacturing methodology. It is this part where contributions to the kinematic geometry of ruled surfaces in contact, differential geometry of surfaces in direct contact, along with toothed bodies in mesh are developed. Part Three includes design formulas to rate or evaluate gear pairs generated using the developed methodology. Practicing gear engineers can bypass the analytical treatment of Parts One and Two and focus on Part Three. Part Three discusses the design procedure based on the analytical development and gives several examples to illustrate the capabilities of the new approach. A noteworthy feature of the developed methodology is that the design and manufacturing data for the toothed bodies that satisfy the stated requirements and the cutters used to produce them are synthesized concurrently and interactively in a PC environment. The synthesized shapes of the gear and cutter elements along with the surfaces of the teeth separately or in conjugate action are displayed graphically. The designer can view and evaluate trial designs prior to further analysis or manufacturing. Sample displays are included as part of the final chapter to illustrate the process in a variety of nonconventional as well as conventional applications. Included are 10 appendices.

All the relations presented in this text have been coded and tested. Delgear©, a computer software package developed by the author, is included as part of this book. The requirements necessary to run Delgear© are standard with PCs and laptops today. This software enables the reader to specify motion (circular and noncircular gears), tooth type (involute and cycloidal), gear type, cutter, and manufacturing parameters and view the results of the integrated CAD–CAM process for generalized gear pairs. The included software is bundled into an install package that prepares a windows-based environment to use the Delgear© package. Installation instructions are provided in Appendix D. A user's guide is included with the Delgear© software to assist its usage. Included are 22 illustrative examples of gear pairs, both traditional and nontraditional gear pairs, to illustrate features presented in this book.

Much of the mathematics used in this book is presented in existing textbooks and is not summarized. However, the novel information of this book is preceded with a level of basic mathematics. Intermediate graphical displays of gear results and equations developed in Chapters 2–6 are deferred to Chapter . The fundamental theory developed in Chapters 2–6 is presented with figures and equations for individuals interested in the kinematic geometry of gear elements. Gearing is not a field of study analogous to mathematics, vibrations, FEA, or fluid mechanics; and consequently, exercises at the end of each chapter are not included in a traditional textbook manner. The dedicated reader can use the Delgear© software package to check intermediate values at each stage of the presented methodology.

The webpage www.wiley.com/go/dooner_2e provides supplementary material to the Kinematic Geometry of Gearing. This webpage provides a link that enables interested readers to freely download and use software developed by the author. The developed software facilitates the geometric design and rating of various gear types including spur, helical, spiral bevel, straight bevel, spur and spiral non-circular gears, spur and spiral hypoid gears, non-orthogonal worm gears, along with nontraditional gear types.

Acknowledgments are of the order to express the author's appreciation for facilitating the presentation in this work. First, an acknowledgment is due to the late Prof. Ali Seireg for his collaboration and encouragement on the original work and sharing the importance of system design. Behind-the-scenes facilitators include John Wiley & Sons, Ltd for their willingness to continue with this second edition along with IBM APL Product and Services for disentangling a variety of programming woes; especially Nancy Wheeler and David Liebtag, formerly of IBM APL Product and Services. An extended acknowledgement goes to Dr. Michael W. Griffis for his role in the presentation of this new gear approach by fielding many questions and providing insight into the theory of screws. And finally, recognition to IMPO for the fortitude and patience to bear with me as I pieced together this manuscript and software.

David Dooner Mayagüez, Puerto Rico

1.http://www-groups.dcs.stand.ac.uk/~history/Mathematicians

Part One

Fundamental Principles of Toothed Bodies in Mesh

Before we can understand the future, we must learn about the past

–Anonymous

1

Introduction to the Kinematics Of Gearing

1.1 Introduction

A brief history of gearing and some established gear concepts are presented in this chapter as an introduction to the development of a generalized kinematic theory for the design and manufacture of gears. The primary objective is to familiarize the kinematician with gear terminology in a format that is familiar to them (compatible with established kinematic theory) as well as to introduce the gear specialist to some of the relevant kinematic concepts that are used in developing a generalized methodology for the concurrent design and manufacture of gear pairs. This approach includes the synthesis and analysis of the gear elements concurrently with the design of the corresponding cutter elements used for their fabrication. These introductory concepts will be built upon throughout this book to develop a generalized methodology based on kinematic geometry for the integrated design and manufacturing of appropriate toothed body to transmit a specified speed and load between generally oriented axes and the constraints that may restrict implementation.

1.2 An Overview

An introduction to the complexities involved in the design and manufacture of toothed bodies in mesh can be achieved by first examining the kinematic structure of conjugate motion between parallel axes. One purpose of this chapter is to introduce the concept of toothed wheels and demonstrate the basic kinematic geometry of toothed wheels in mesh as well as their fabrication. This extended introduction is intended to establish a foundation that will be used as a corollary to exemplify the intricacies of spatial gearing (namely, worm and hypoid gearing). A similar introductory treatment on gears is presented in existing textbooks on kinematics and machine design (e.g., Spotts, 1964; Martin, 1969; Shigley and Uicker, 1980; Erdman and Sandor, 1997; Budynas and Nisbett, 2011). The elementary treatment provided in these textbooks on kinematics and machine design is essentially based on the books by Buckingham 1949 and Merritt (1971). Because of its practical importance, the design and manufacture of toothed bodies continues to attract the attention of researchers in a variety of fields (e.g., geometry, lubrication, dynamics, elasticity, material science, and computer science). Dudley (1969) provides a brief account on the history of gears, and additional information regarding the history of gears is provided by Cromwell (1884) and Grant (1899). An overview on the design and manufacture of gears is presented by Dudley (1984) and Drago (1988). Specialists in the gear industry have contributed to the second edition of Dudley's Gear Handbook edited by Townsend (1991). A more extensive and up-to-date analysis for the design and manufacture of gears is provided by the following organizations:

American Gear Manufacturers Association (AGMA)International Standards Organization (ISO)Deutsches Institute für Normung (DIN)Japanese Gear Manufacturers Association (JGMA)American National Standards Institute (ANSI)British Gear Association (BGA)

One of the earliest documented geared devices is the South Pointing Chariot. A model of a South Pointing Chariot is depicted in Figure 1.1. The function of this device is to serve as a mechanical compass in crossing the Gobi dessert. The statue atop of the wheeled cart maintains a constant direction of pointing independent of the cart track. Various claims to the date of the device range from 2700 BC to 300 AD. Heron of Alexandria devised many mechanical systems involving mechanisms (some geared). Example systems include special temple gates, mechanized plays, coin-operated water dispensers, and the aeolipile. Leonardo da Vinci is one of the most celebrated designers of all times. Da Vinci is credited with the various sketching of gears in Figure 1.2.

Figure 1.1 South pointing chariot (reproduced by permission of Science Museum London/Science and Society Picture Library)

Figure 1.2 Gear sketches by da Vinci (reproduced by permission of Biblioteca Nacional)

Norton (2001) credits James Watt as the “first” kinematician for documenting the coupler motion of a four-link mechanism. This documentation was part of his effort to achieve long strokes on his steam engine. More noted is Euler (father of involute gearing) and his analytical treatment of mechanisms. Yet, Reuleaux is considered the “father” of modern kinematics for his text . Reuleaux defined six basic mechanical components (namely, a link, wheel, cam, screw, ratchet, and belt). A gear can be considered a manifestation of the wheel, cam, and screw.